For some research I'm doing for my PhD, I encountered this situation where I'm pretty sure what I'm doing is correct, but I'd like some confirmation from someone with more knowledge of this stuff. The situation can be described in a very general way as follows:
Suppose we have a 3xN (N > 3) matrix M with any three columns of M being linearly independent, an Nx1 vector U where each entry is strictly positive, and let M*U = P.
Suppose further that we perturb M by $\epsilon$ in some way, to $M_\epsilon$. If $\epsilon$ is small enough, $M_\epsilon$ is still full rank, and thus there are infinitely many solutions $U_\epsilon$ to $M_\epsilon$ * $U_\epsilon$ = P for each $\epsilon$. What I would like to do is say "For every $\epsilon$ > 0, define $U_\epsilon$ to be the unique solution to $M_\epsilon$ * $U_\epsilon$ = P such that ||$U_\epsilon$ - U|| is minimal (and there's no reason to assume that's unique, so I would have to maybe specify a further condition, like minimal under a lexicographical ordering or something?).
Once $U_\epsilon$ is defined in this way, then I want to say that it converges to U as $\epsilon$ goes to 0, and therefore we can choose a small enough $\epsilon$ so that all entries of $U_\epsilon$ are positive (this is the main goal). The main problem I have is that without trying to specify a unique $U_\epsilon$, all I would know is that the $U_\epsilon$'s would converge to some solution of M*X = P, but not necessarily the U that I want, which I need to prove that $U_\epsilon$ can have all positive entries. But I'm not 100% sure the way that I've specified $U_\epsilon$ is reasonable or gives me the result that I want.
Hopefully the way I have explained the problem and what I'm looking for is clear enough, if not just let me know. Figuring this out will be very helpful for my PhD research.
The solutions to $M_\epsilon X = P$ form an affine subspace $S$ of $\mathbb R^n$, and in particular a convex set. If $\| \cdot \|$ is the usual Euclidean norm, since that norm is strictly convex there is always a unique member of $S$ that minimizes $\|X - U\|$.
EDIT: Since $M$ has rank $3$, there is a set of three columns which are linearly independent and thus form a nonsingular $3 \times 3$ matrix. WLOG let's say these are the first three columns and write $M$ in block form as $[B \mid C]$, where $B$ is $3 \times 3$, and correspondingly $U = \pmatrix{V \cr \hline W}$. Note that $M U = P$ is equivalent to $V = B^{-1}(P - C W)$.
Now if $M_\epsilon = [B_\epsilon \mid C_\epsilon]$ is sufficiently close to $M$, $B_\epsilon$ will still be nonsingular and we have a solution $\tilde{U}_\epsilon = \pmatrix{B_\epsilon^{-1}(P - C_\epsilon W)\cr \hline W}$. Since inversion is a continuous map on the nonsingular matrices, $\|\tilde{U}_\epsilon - U\| \to 0$ as $\epsilon \to 0$. Since $\|U_\epsilon - U\| \le \|\tilde{U}_\epsilon - U\|$, that also goes to $0$ as $\epsilon \to 0$. In particular, if $\epsilon$ is sufficiently small all entries of $U_\epsilon$ must be strictly positive.